Otimal Design of Pressure Relief Valves in Hydroower Stations Jianxu Zhou 1, Bryan W. Karney 2, Fulin Cai 1 1 College of Water Conservancy and Hydroower Engineering, Hohai Univ., Nanjing 2198, Jiangsu, China 2 Det. of Civil Engineering, Univ. of Toronto, Toronto M5S 1A4, Ontario, Canada Abstract For some hydroower stations with longer ressurized ielines, the by-ass ressure relief valves (PRVs) with an otimized coordinating control strategy are referred to surge tanks for controlling transient ressures. Based on the method of characteristics, a simulation model of a hydro-mechanical system is built to connect the detailed flow characteristics of the PRV to a coordinating control strategy for the wicket gate closing law. This overall model must consider whether or not the PRV normally oens as well as the nature of both its oening and closing relations. Furthermore, the SMS formula for the maximum rising rate of unit s rotational seed is discussed in detail and included in further analytical analysis. Based on this referred emirical formula, for the maximum rising rate of unit s rotational seed, the consistency and deviation between its analytical solutions and numerical solutions are further analyzed and illustrated. Secific system analysis also indicates that, with a sufficient flow caacity of the PRVs and an otimal coordinating control strategy for the units and PRVs, all the oerational constraints including the maximum water hammer and maximum unit s rotational seed can be satisfied. Keywords: coordinating control strategy; hydraulic transient; hydroower; ressure relief valve Introduction With the growing utilization of water resources, many hydroower stations with relatively low head and long ressurized ielines have been built or exanded in many countries. For these hydroower stations, the oerating stability is relatively easy to guarantee considering the otimal control of the governor and the favorable effects of the ower system; however, more attention should be aid to the detailed transient analysis of the hydro-mechanical system and the exected solutions should also be satisfied with transient control strategy. Otherwise, necessary mitigations must be considered to achieve effective transient control. The surge tank with a reasonable and effective area has often been the rotection system of choice to mitigate the ressure surge and imrove the oerating stability of the hydroower stations as long as suitable geological and toograhic conditions for its layout are available; but, sometimes the surge tanks are uneconomical, esecially for the small hydroower stations with relatively low head and long ressurized ielines. If an ustream surge tank is to be avoided, the ossible alternative mitigations mainly include the by-ass ressure relief valve (PRV), delayed wicket closing times, otimized wicket closing law, an increased rotational inertia GD 2, use of safety membranes or an increased diameter of enstock (Liu et al., 21). Yang et al. (1999) discussed the main technical measures and their alication instead of the ustream surge tank to imrove the hydraulic transients in small ower stations. Hu et al. (1989) and Ni et al. (1996) introduced the design and mechanism of the safety membrane to control the maximum water Page 1 of 9
hammer, sometimes combined with PRVs. Lü et al. (27) and Wang et al. (21) analyzed the feasibility to cancel the ustream surge tank of two hydroower stations and suggested an otimized wicket closing law along with reasonably increased diameter of enstock instead of the by-ass PRVs. Based on the detailed comarison of hydraulic transients with/without PRVs in a given hydroower station, Zhu (25) roosed a reasonable alternative for the hydro-mechanical system that avoids PRVs. If a PRV is avoided, a safety or ruture membrane is required for rotection against in load rejection but its design is relatively comlex and requires some comlementary infrastructure. Using an otimized wicket closing law is challenging since it does not benefit from other measures and additional equiment. In some cases, the transients control strategy can be realized with an otimized wicket closing law and roerly increased diameter of enstock or units rotational inertia GD 2, otherwise, a by-ass PRV with necessary ieline for each unit should be designed instead of the surge tank. Some hydroower stations with by-ass PRVs have been aralleled in oeration or are in construction (Zhou et al., 29). PRVs have been widely used in water suly systems. Douglas (21) analyzed some devices including the PRV to achieve the surge control in the uming system. Jung et al. (26) focused on the otimization of the hydraulic devices in a ieline system to control its transient resonse, esecially for PRVs. Zhang et al. (28) strengthened that an aroriately designed PRV would rotect some systems from excessive water hammer effectively for the uming system. Meanwhile, the by-ass PRVs also began to be alied in the umstorage ower stations for transient ressure control successfully (Adamkowski, 199). The PRV is a commonly used mitigation for transient ressure control, esecially in the hydroower stations with relatively low head and long ressurized ielines, which have no strict requirement for the electric energy quality and will not undertake frequency regulation for the ower system. In such cases, the PRV makes basically no contribution to the system s stability and regulation erformance. In order to suress the water hammer, a by-ass PRV is ositioned near the inlet of the siral case of each unit, and the coordinating control strategy is selected based on hydraulic transient analysis and considerations. Simulation models and investigation of coordinating control strategy For the hydroower stations with the PRVs, the method of characteristics (Wylie et al., 1993) is also used for hydraulic transients comutation, only with an additional boundary condition PRV, showed in Fig.1. C + n 1 Fig. 1. Schematic of a PRV n 1 C Pressure relief valve In order to realize valid suression of the water hammer, the reasonable flow matching between the PRV and the turbine is crucial. With this remise, if a secified PRV has been designed for a given hydroower station, its flow is only relevant to the oening and the dynamic head. For the water flow in the ustream and downstream ies of the PRV, the characteristic equations are C + : H = C B Q (1) n C : H 1 = Cm + BmQ 1 (2) n Page 2 of 9
where C, B, C m and B m are coefficients associated with the ie characteristics, and the known iezometric heads and flows of the adjacent sections at t- t; H n and Q n are unknown iezometric head and flow of section n at t for ustream ie; H 1 and Q 1 are unknown iezometric head and flow of section 1 at t for downstream ie. The flow of the PRV, Q V, is satisfied with Q = Q = Q (3) V n The dynamic head of the PRV, H V, the ressure difference acting on the PRV, is H = H H (4) V n Relative to different PRV s oening y, the flow characteristic of PRV is resented as QV = f( HV, y) (5) Based on the above boundary conditions of PRVs, combined with the characteristic equations of ressurized ielines and other necessary boundary conditions (Wylie et al., 1993), the simulation model can be built for further hydraulic transient analysis. Considering a hydroower station with a by-ass PRV for each unit, as the load rejection haens, the closure of wicket gate is immediately initiated to reduce the rate of rise of the unit s rotational seed, and the PRV coordinately controlled by the same governor is synchronously oened to discharge the flow to control the water hammer. After wicket gate is fully closed, a closing action should also be initiated for the PRV at a certain time to avoid extra waste of water and energy, and in this stage, it is also necessary to confirm the maximum water hammer not exceeding the allowable value. Therefore, a reasonable coordinated control strategy of the unit and the PRV should be determined on the basis of the allowable rising rate of rotational seed and water hammer. On the other hand, if the PRV is inactive for load rejection, wicket gate should slowly be closed to suress the 1 1 water hammer, and in this case, the maximum rotational seed should be controlled to be less than the maximum runaway rotational seed. In addition, the PRVs are inactive for load accetance and relatively minor load fluctuations (less than 15%), so they will not contribute to the oerating stability of the hydroower stations. Transient ressure control during load accetance and stability analysis should be carried out without regard to the PRVs. As a straight-line closing law is otimized for wicket gate, the oening time of the PRV is often equal to the closing time of the unit. On the other hand, if a broken-line closing law is required for wicket gate, the break would be better to be close to the idle oerating oint, and then, PRV s oening time is basically equal to unit s first-stage closing time. The main advantage of a broken-line closing law for the wicket gate is to reduce the required flow caacity of the PRV (using a relatively small diameter or short stroke) but still to satisfy the transient control requirements. Unfortunately, the coordination is still comaratively comlex. Analytical analysis of the rising rate of unit s rotational seed Usually, with the rolonging of wicket closing time, the maximum rising rate of rotational seed gradually increases. For most of the commonly used analytical formulas (Lin, 1995; Yang et al., 27), it tends inevitably to infinity or a constant if wicket closing time is sufficiently long. Generally, both the fast-closing time and the slow-closing time are designed for the hydroower stations with the by-ass PRVs, and then, the maximum rising rate of rotational seed increases with the rolonging of wicket fast-closing time as the PRVs are normally oened, on the other hand, it will ossibly be reduced by the Page 3 of 9
rolonging of wicket slow-closing time considering the effect of the runaway characteristics of the turbines as the PRVs are inactive. Actually, this effect is often ignored in many emirical formulas. Therefore, it is necessary to fully reveal the change trend of the rising rate of rotational seed in a wider time-domain for the hydroower stations with a PRV for each unit (Zhou et al., 29). Considering the straight-line closing law, on the basis of analytical analysis of several emirical formulas, the SMS formula (Lin, 1995) is referred which comrises the effects of the water hammer and unit s runaway characteristics. β = β y cf (6) where c= 1+ β y 1 ( n n 1), is the influence coefficient of unit s runaway 2 2 characteristic, β y = 182.5N T s 1 ( n GD ) ; 1.5 f = ( 1+ ξ ), is the influence coefficient of water hammer; ξ is the rising rate of the water hammer; T s1 is wicket valid closure time, T s1 =kt s ; n and n are unit s runaway seed and initial seed resectively; T s is wicket closing time; N is unit s initial outut; GD 2 is unit s rotational inertia. Introducing unit s inertia time 2 2 constant T a = GD n 365N ), and defining ( k = n n, equation (6) simlifies to k( k 1) Ts 1. 5 β = β f = (1+ ξ ) (7) 2( k 1) T + kt a As T s,β in (7) for limit tends to be limβ = k 1 (8) T s This final form indicates that, if the effect of water hammer is not considered temorarily, with the rolonging of wicket closing time, the maximum rising rate of rotational seed gradually increases. If s wicket closing time tends to be infinite, it will aroximately be a constant. When the effect of water hammer is considered, with the rolonging of wicket closing time, water hammer is mitigated, and then f will gradually decrease. For the hydroower stations without PRVs, the trend also becomes gentle for relatively longer closing time. By contrast, for the hydroower stations with PRVs, as a wicket slow-closing law is required for inactive PRVs, the lowering of f is relatively obvious, while β tends to be a constant with relatively long wicket closing time. Therefore, β can slightly decrease with the rolonging of wicket closing time in this case. On the basis of the above analysis, SMS formula reasonably reflects the effects of water hammer and unit s runaway characteristics, and actually reveals the change trend of unit s rotational seed in the hydroower stations with PRVs. Therefore, it is used for further confirmation with numerical simulation. Numerical results with case analysis Reservoir Pressure relief valve Unit 1 Unit 2 Tail water Fig. 2. Layout of a hydroower station with PRVs Fig. 2 illustrates a two turbines system with a longer ressurized enstock. A by-ass PRV with necessary ielines is equied for each unit, and its designed maximum flow at full oening is 47.35 m 3 /s with the maximum ustream ressure 6. m and its allowable relative travel is.8 to reserve some extra flow caacity for safe oeration. Page 4 of 9
The allowable values for the hydraulic transients control include the allowable maximum rising rate of unit s rotational seed, 5%, and the allowable maximum internal water ressure at the inlet of siral case, 5. m. Esecially, as PRVs are occasionally inactive, the maximum rising rate of unit s rotational seed should not exceed 11.3%. The lag time of the governor has been taken as.15s. On the basis of the given data for the hydro-mechanical system, the arameters in (7) for the worst case can be calculated out including T a =6.35s, k =1.69 and k=.8. Because the analytical formulas for different water hammer calculation are unsuitable for the coordinating regulation of PRVs, after the numerical solutions of maximum internal water ressure at the inlet of the siral case H C is comuted out, its relative increment, ξ c, are referred for ξ in (7). Fig. 3 intuitively shows the change trends of β and f for wicket slow-closing time. β, f 2. 1.5 1..5. f β 7 75 8 85 9 95 T s (s) Fig. 3. β t relation and f t relation for wicket slow-closing time Fig. 3 shows the same trends for β and f resectively as analyzed above when wicket slow-closing time is used for inactive PRVs. Subsequently, the analytical solutions β ana are calculated by SMS formula for different wicket closing time in Tables 1 and 2 along with its numerical solutions β num for further comarative analysis. Table. 1. β for wicket fast-closing time T S (s) H c (m) ξ c β num β ana 1 5.27.416.435.555 11 49.45.393.454.568 12 48.74.373.471.58 13 47.82.347.488.585 14 47.14.328.53.593 15 46.47.39.516.597 Table. 2. β for wicket slow-closing time T S (s) H c (m) ξ c β num β ana 7 5.94.435.989 1.25 75 49.91.46.979 1.3 8 49.2.386.97.99 85 48.24.359.96.968 9 47.39.335.951.949 95 46.58.312.941.93 Several imortant results follow from this. First, as wicket slow-closing law is designed for inactive PRVs, the change trend of β has obvious difference from that for wicket fast-closing law and it slightly decreases with the rolonging of wicket slow-closing time. These oosite change trends for wicket fast-closing law and wicket slow-closing law agree well with the analytical solutions based on the SMS formula. Meanwhile, the deviation between analytical solutions and numerical solutions of β is relatively large as PRV s normal oening, which mainly deends on the comlex water hammer characteristics with coordinating control of the turbines and PRVs. As the PRVs are inactive, because the water hammer characteristics have no relation with the flow characteristics of the PRVs, this deviation is tiny. Therefore, as PRV s normal oening, the emirical formulas including the SMS formula have lower recision for analytical solutions of β for the hydroower stations with PRVs. Considering PRV s normal oening for load rejection, wicket fast-closing law is Page 5 of 9
referred for the turbine to reduce the maximum rising rate of rotational seed, and the PRV simultaneously oens to discharge flow in order to suress the water hammer. Base on the worst case, wicket fast-closing law and PRV coordinating oening law are further analyzed in detail. With the reliminary analysis, though the required flow caacity of the PRV for wicket straightline closing law is relatively large, it is also referred for further otimal analysis in this case because of its reliable coordinating control and the accetable flow increment. Because the errors are relatively large for the analytical solutions mentioned above, the otional wicket linear fast-closing time is 11-13 s on the basis of the numerical solutions in Table 1. Considering reasonable allowances, the otimized closing time is 13s, which is also PRVs linear oening time. For load accetance without regard to PRVs, wicket oening time deends on the allowable minimum internal water ressure along the ressurized enstock. If two turbines accet the full load at the same time, wicket oening time T s should not be less than 52. s. In reality, because oerational control is ossible for load accetance, the worst case is usually the successive load accetance over reasonable time interval. Therefore, for load accetance of one turbine, the minimal wicket oening time T s is shortened to 2. s to meet the requirement of minimum water hammer control. As an emergency control for inactive PRVs, a slow-closing wicket law is a viable alternative. In this case, the maximum internal water ressure should also be lower than the allowable value and the maximum rotational seed should not exceed the secified runaway seed. The above analysis shows that, with the rolonging of wicket slow-closing time, the maximum rising rate of rotational seed slightly decreases with the gradual decreasing maximum water hammer. On the basis of both numerical solutions and analytical solutions in Table 2, wicket slow-closing time should not be less than 75 s and then the referred closing time is 85 s considering the immediate recommissioning in this emergency case. As the PRV fully oens along with wicket gate full closing, the PRV should be closed according to a reasonable closing law as soon as ossible to reduce loss of water and energy. In this stage, the maximum internal water ressure should also be controlled. Generally, the PRV begins to close from full oening after a time interval considering governors erformance. In reality, if ossible, the PRV is immediately closed as full oening is realized. Based on this consideration, PRV immediate closing time is 96 s as it is fully oened in 13.15 s including the lag time of the governor. Based on the above otimal analysis, the coordinating control strategy is shown in Fig. 4, in which all the oenings of the turbine and the PRV are resented by relative values. y 1.8.6.4.2 (.15, 1.) (13.15,.8) (13.15, ) (85.15, ) (19.15, ) 25 5 75 1 125 t (s) Wicket fast-closing law Wicket slow-closing law PLV oening-closing law Fig.4. Coordinating control strategy According to the above coordinating control strategy, Fig. 5 illustrates the dynamic curves of internal water ressure at Page 6 of 9
the inlet of the siral case H C and PRV s flow Q V. HC (m), QV (m 3 /s) 5 4 3 2 1 Q V H C 3 6 9 12 15 t (s) Fig.5. Dynamic curves of H C and Q V It can be shown that, the maximum H C is less than the allowable value 5. m, and the maximum PRV s flow is less than 47.35m 3 /s with a relatively low ustream ressure. These all meet the requirement of transient ressure control and the flow caacity of the recommended PRV is also accetable with a suitable safety allowance. Finally, the stability analysis under small disturbance is carried out without regard to PRVs by means of state equations and eigenvalues analysis (Wylie et al., 1993). On the basis of the worst case, introducing governor s arameters including daming time constant T d and temorary seed dro b t, the stable regions of small disturbance are obtained for full-load oeration, art-load oeration and no-load oeration resectively, as showed in Fig. 6. bt 1..8.6.4.2. Full-load Part-load No-load Unstable region 2 4 6 8 1 T d (s) Stable region Fig. 6. Stable regions of small disturbance The results indicate that, for the hydroower stations with PRVs, because of the relatively large water inertia time constant of the enstock, the stable regions are comaratively small; if the governor s arameters are determined according to the obtained stable regions, the hydromechanical system can satisfy the requirement of oerating stability. Conclusions Recently, in order to realize transient ressure control, the by-ass PRV with necessary ieline is often used instead of the surge tank or other mitigations in the hydroower stations with relatively low head and long ressurized ielines. As a traditional analytical formula for the maximum rising rate of rotational seed, the SMS formula nicely illustrates the key trends associated with different hydroower stations. For the hydroower stations with PRVs, in order to control the transient ressure, wicket fast-closing law is required for a PRVs normal oening, while wicket slow-closing law is for inactive PRVs. The rising rate of unit s rotational seed usually increases with the rolonging of wicket closing time; on the other hand, for wicket slow-closing law, it slightly decreases because of the effects of water hammer and unit s runaway characteristics. With a given hydroower station with PRVs, based on the SMS formula, the numerical solutions agree well with the analytical solutions, esecially for wicket slow-closing law. Furthermore, hydraulic transient control can be realized with a reasonable coordinating control strategy, and the stable regions of small disturbance are comaratively small without the contribution of PRVs. References Page 7 of 9
Adamkowski A. (199). Alication of the by-ass flow control for water hammer suression in um-turbine installation. Symosium on Hydraulic Machinery and Cavitation, Modern Technology in Hydraulic Energy Production, IAHR, vol. 1, 11-14. Douglas J. F., Gasiorek J. M. and Swaffield J. A. (21). Fluid Mechanics, 4th edition, Prentice Hall, Pearson Education Limited, England. Hu P. C., Zheng P. S. and Elkouh A. F. (1989). Relief valve and safety membrane arrangement in lieu of surge tank. Journal of Energy Engineering, ASCE, 115(2), 78-83. Jung B. S. and Karney B. W. (26). Hydraulic otimization of transient rotection devices using GA and PSO aroaches. Journal of Water Resources Planning and Management, ASCE, 132(1), 44-52. Liu Q. Z. and Hu M. (21). Hydroower stations, China Water Power Press, Beijing, China. Lin Y. Y. (1995). Turbine regulation and auxiliary equiment, China Water Power Press Beijing, China. Lü K. and Feng S. N. (27). Design of the cancellation of ustream surge tank in the water diversion and ower generation system of Zhougongzhai Project. Zhejiang Hydrotechnic, (1), 34-36. Ni F. S., Hu P. C. and Wang Q. B. (1996). Numerical simulation of hydraulic transients in hydroower lant using safety membranes. Journal of hydraulic engineering, ASCE, 122(6), 298-3. Wang D. C., Wang J. and Shi L. H. (21). Design and investigation of the cancellation of surge tank of the Heishan hydroower station. Heilongjiang Science and Technology of Water Conservancy, 38(5), 45-46. Wylie E. B., Streeter V. L. and Suo L. S. (1993). Fluid transients in systems, Prentice Hall Englewood Cliffs, New Jersey, American. Yang J. D., Zhang J. J. and Jiang Q. (27). Research on the influencing factors of rising ratio of rotating seed. Journal of Hydroelectric Engineering, 26(2): 147-152. Yang J. G. and He W. X. (1999). Analysis of the main technical measures of regulation guarantee in relacing surge tank. Journal of Northwestern Agricultural University, 27(6), 117-119. Zhang Q. F., Karney B. W. and McPherson D. L. (28). Pressure relief valves selection and transient ressure control. Journal of American Water Works Association. 1(8), 62-69. Zhou J. X., Cai F. L. and Huang Y. G. (29). Analysis of rising ratio of rotating seed in hydroower stations with ressure regulating valve. Journal of Hydroelectric Engineering, 28(6): 214-218. Zhu X. (25). Research on hydraulic transient rocess of Gudongkou hydroower station. Hubei Water Power, (4), 21-24. Acknowledgments The aer was comleted within the research rojects funded by the National Natural Science Foundation of China under Grant Nos. 96127 and 517951, and also financially suorted by the Fundamental Research Funds for the Central Universities of China and the Priority Academic Program Develoment of Jiangsu Higher Education Institutions (PAPD). Disclosures Authors have nothing to disclose. Page 8 of 9
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